1. If a1 = λ1i + µ1j + ν1k, a2 = λ2i + µ2j + ν2k, a3 = λ3i + µ3j + ν3k, where {i,j, k} is a standard basis, show
that
(a)
a1 · (a2 × a3) =
λ1 µ1 ν1
λ2 µ2 ν2
λ3 µ3 ν3
.
(b) Deduce that a2 · (a3 × a1), due to cyclic rotation of the vectors in a triple scalar product leaves the
value of the product unchanged.
(c) If r(t) = (3t
2 − 4)i + t
3
j + (t + 3)k, where {i,j, k} is a constant standard basis, find r˙ and ¨r. Deduce
the time derivative of r × r˙.